A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES

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1 Theory and Applications o Categories, Vol. 24, No. 21, 2010, pp A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN Abstract. Notions o generalized multicategory have been deined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are deined as the lax algebras or Kleisli monoids relative to a monad on a bicategory. However, the meanings o these words dier rom author to author, as do the speciic bicategories considered. We propose a uniied ramework: by working with monads on double categories and related structures (rather than bicategories), one can deine generalized multicategories in a way that uniies all previous examples, while at the same time simpliying and clariying much o the theory. Contents 1 Introduction Virtual double categories Monads on a virtual double category Generalized multicategories Composites and units categories o T -monoids Virtual equipments Normalization Representability 628 A Composites in Mod and H-Kl 636 Comparisons to previous theories Introduction A multicategory is a generalization o a category, in which the domain o a morphism, rather than being a single object, can be a inite list o objects. A prototypical example is the multicategory Vect o vector spaces, in which a morphism (V 1,..., V n ) W is a The irst author was supported by a PIMS Calgary postdoctoral ellowship, and the second author by a National Science Foundation postdoctoral ellowship during the writing o this paper. Received by the editors and, in revised orm, Transmitted by Tom Leinster. Published on , this revision Mathematics Subject Classiication: 18D05,18D20,18D50. Key words and phrases: Enriched categories, change o base, monoidal categories, double categories, multicategories, operads, monads. c G.S.H. Cruttwell and Michael A. Shulman, Permission to copy or private use granted. 580

2 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 581 multilinear map. In act, any monoidal category gives a multicategory in a canonical way, where the morphisms (V 1,..., V n ) W are the ordinary morphisms V 1... V n W. The multicategory Vect can be seen as arising in this way, but it is also natural to view its multicategory structure as more basic, with the tensor product then characterized as a representing object or multimorphisms. This is also the case or many other multicategories; in act, monoidal categories can be identiied with multicategories satisying a certain representability property (see [Her00] and 9). In addition to providing an abstract ormalization o the passage rom multilinear map to tensor product, multicategories provide a convenient way to present certain types o initary algebraic theories (speciically, strongly regular initary theories, whose axioms involve no duplication, omission, or permutation o variables). Namely, the objects o the multicategories are the sorts o the theory, and each morphism (X 1,..., X n ) Y represents an algebraic operation o the theory. When viewed in this light, multicategories (especially those with one object, which correspond to one-sorted theories) are oten called non-symmetric operads (see [May72]). The original deinition o multicategories in [Lam69] (see also [Lam89]) was also along these lines (a ramework or sequent calculus). The two viewpoints are related by the observation that when A is a small multicategory representing an algebraic theory, and C is a large multicategory such as Vect, a model o the theory A in C is simply a unctor o multicategories A C. This is a version o the unctorial semantics o [Law63]. Our concern in this paper is with generalized multicategories, a well-known idea which generalizes the basic notion in two ways. Firstly, one allows a change o base context, thereby including both internal multicategories and enriched multicategories. Secondly, and more interestingly, one allows the inite lists o objects serving as the domains o morphisms to be replaced by something else. From the irst point o view, this is desirable since there are many other contexts in which one would like to analyze the relationship between structures with coherence axioms (such as monoidal categories) and structures with universal or representability properties. From the second point o view, it is desirable since not all algebraic theories are strongly regular. For example, generalized multicategories include symmetric multicategories, in which the inite lists can be arbitrarily permuted. Representable symmetric multicategories correspond to symmetric monoidal categories. Enriched symmetric multicategories with one object can be identiied with the operads o [May72, Kel05, KM95]. These describe algebraic theories in whose axioms variables can be permuted (but not duplicated or omitted). In most applications o operads (see [EM06, M03] or some recent ones), both symmetry and enrichment are essential. An obvious variation o symmetric multicategories is braided multicategories. I we allow duplication and omission in addition to permutation o inputs, we obtain (multisorted) Lawvere theories [Law63]; a slight modiication also produces the clubs o [Kel72b, Kel92]. There are also important generalizations to algebraic theories on more complicated objects; or instance, the globular operads o [at98, Lei04] describe a certain sort o algebraic theory on globular sets that includes many notions o weak n-category.

3 582 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN A very dierent route to generalized multicategories begins with the observation o [ar70] that topological spaces can be viewed as a type o generalized multicategory, when inite lists o objects are replaced by ultrailters, and morphisms are replaced by a convergence relation. Many other sorts o topological structures, such as metric spaces, closure spaces, uniorm spaces, and approach spaces, can also be seen as generalized multicategories; see [Law02, CT03, CHT04]. With so many dierent aces, it is not surprising that generalized multicategories have been independently considered by many authors. They were irst studied in generality by [ur71], but have also been considered by many other authors, including [Lei04], [Lei02], [Her01], [CT03], [CHT04], [ar70], [Web05], [D98], [Che04], and [DS03]. While all these authors are clearly doing the same thing rom an intuitive standpoint, they work in dierent rameworks at dierent levels o generality, making the ormal deinitions diicult to compare. Moreover, all o these approaches share a certain ad hoc quality, which, given the naturalness and importance o the notion, ought to be avoidable. In each case, the authors observe that the something else serving as the domain o morphisms in a generalized multicategory should be speciied by some sort o monad, invariably denoted T. For example, ordinary multicategories appear when T is the ree monoid monad, globular operads appear when T is the ree strict ω-category monad, and topological spaces appear when T is the ultrailter monad. All the diiculties then center around what sort o thing T is a monad on. Leinster [Lei02, Lei04] takes it to be a cartesian monad on an ordinary category C, i.e. C has pullbacks, T preserves them, and the naturality squares or its unit and multiplication are pullback squares. urroni [ur71], whose approach is basically the same, is able to deal with any monad on a category with pullbacks. Hermida [Her01] works with a cartesian 2-monad on a suitable 2-category. arr and Clementino et. al. [ar70, CT03, CHT04] work with a monad on Set equipped with a lax extension to the bicategory o matrices in some monoidal category. Weber [Web05] works with a monoidal pseudo algebra or a 2-monad on a suitable 2-category. aez-dolan [D98] and Cheng [Che04] (see also [FGHW08]) use a monad on Cat extended to the bicategory o prounctors (although they consider only the ree symmetric strict monoidal category monad). Inspecting these various deinitions and looking or commonalities, we observe that in all cases, the monads involved naturally live on a bicategory, be it a bicategory o spans (urroni, Leinster), two-sided ibrations (Hermida), relations (arr), matrices (Clementino et. al., Weber), or prounctors (aez-dolan, Cheng). What causes problems is that the monads o interest are requently lax (preserving composition only up to a noninvertible transormation), but there is no obvious general notion o lax monad on a bicategory, since there is no good 2-category (or even tricategory) o bicategories that contains lax or oplax unctors. Furthermore, merely knowing the bicategorical monad (however one chooses to ormalize this) is insuicient or the theory, and in particular or the deinition o unctors and transormations between generalized multicategories. Leinster, urroni, Weber, and Hermida can avoid this problem because their bicategorical monads are induced by monads

4 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 583 on some underlying category or 2-category. Others resolve it by working with an extension o a given monad on Set or Cat to the bicategory o matrices or prounctors, rather than merely the bicategorical monad itsel. However, the various deinitions o such extensions are tricky to compare and have an ad hoc lavor. Our goal in this paper (and its sequels) is to give a common ramework which includes all previous approaches to generalized multicategories, and thereore provides a natural context in which to compare them. To do this, instead o considering monads on bicategories, we instead consider monads on types o double categories. This essentially solves both problems mentioned above: on the one hand there is a perectly good 2-category o double categories and lax unctors (allowing us to deine monads on a double category), and on the other hand the vertical arrows o the double categories (such as morphisms in the cartesian category C, unctions in Set, or unctors between categories) provide the missing data with which to deine unctors and transormations o generalized multicategories. The types o double categories we use are neither strict or pseudo double categories, but instead an even weaker notion, or the ollowing reason. An important intermediate step in the deinition o generalized multicategories is the horizontal Kleisli construction o a monad T, whose (horizontal) arrows X Y are arrows X T Y. Without strong assumptions on T, such arrows cannot be composed associatively, and hence the horizontal Kleisli construction does not give a pseudo double category or bicategory. It does, however, give a weaker structure, which we call a virtual double category. Intuitively, virtual double categories generalize pseudo double categories in the same way that multicategories generalize monoidal categories. There is no longer a horizontal composition operation, but we have cells o shapes such as the ollowing: p 1 X 0 p 2 X X11 p 3 X X2 2 p n X Xn n Y 0 α Y 0 Y 1 q We will give an explicit deinition in 2. Virtual double categories have been studied by [ur71] under the name o multicatégories and by [Lei04] under the name o cmulticategories, both o whom additionally described a special case o the horizontal Kleisli construction. They are, in act, the generalized multicategories relative to the ree category or ree double category monad (depending on whether one works with spans or prounctors). In [DPP06] virtual double categories were called lax double categories, but we believe that name belongs properly to lax algebras or the 2-monad whose strict algebras are double categories. (We will see in Example 9.7 that oplax double categories in this 2-monadically correct sense can be identiied with a restricted class o virtual double categories.) Next, in 3 4 we will show that or any monad T on a virtual double category X, one can deine a notion which we call a T -monoid. In act, we will construct an entire new virtual double category KMod(X, T ) whose objects are T -monoids, by composing g Y 1

5 584 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN the horizontal Kleisli construction mentioned above with the monoids and bimodules construction Mod described in [Lei04, 5.3], which can be applied to any virtual double category. Then in 6 we will construct rom KMod(X, T ) a 2-category KMon(X, T ) o T - monoids, T -monoid unctors, and transormations, generalizing the analogous deinition in [Lei04, 5.3]. This requires a notion o when a virtual double category has units, which we deine in 5 along with the parallel notion o when it has composites. (These deinitions generalize those o [Her00] and can also be ound in [DPP06]; they are also a particular case o the representability o [Her01] and our 9.) For particular X and T, the notion o T -monoid specializes to several previous deinitions o generalized multicategories. For example, i X consists o objects and spans in a cartesian category C and T is induced rom a monad on C, we recover the deinitions o Leinster, Kelly, and urroni. And i X consists o sets and matrices enriched over some monoidal category V and T is a canonical extension o a taut set-monad to X, then we recover the deinitions o Clementino et. al. However, the other deinitions o generalized multicategory cannot quite be identiied with T -monoids or any T, but rather with only a restricted class o them. For instance, i X consists o categories and prounctors, and T extends the ree symmetric monoidal category monad on Cat (this is the situation o aez-dolan and Cheng), then T -monoids are not quite the same as ordinary symmetric multicategories. Rather, a T -monoid or this T consists o a category A, a symmetric multicategory M, and a bijective-on-objects unctor rom A to the underlying ordinary category o M. There are two ways to restrict the class o such T -monoids to obtain a notion equivalent to ordinary symmetric multicategories: we can require A to be a discrete category (so that it is simply the set o objects o M), or we can require the unctor to also be ully aithul (so that A is simply the underlying ordinary category o M). We call the irst type o T -monoid object-discrete and the second type normalized. In order to achieve a ull uniication, thereore, we must give general deinitions o these classes o T -monoid and account or their relationship. It turns out that this requires additional structure on our virtual double categories: we need to assume that horizontal arrows can be restricted along vertical ones, in a sense made precise in 7. Pseudo double categories with this property were called ramed bicategories in [Shu08], where they were also shown to be equivalent to the proarrow equipments o [Woo82] (see also [Ver92]). Accordingly, i a virtual double category X has this property, as well as all units, we call it a virtual equipment. Our irst result in 8, then, is that i T is a well-behaved monad on a virtual equipment, object-discrete and normalized T -monoids are equivalent. However, normalized T -monoids are deined more generally than object-discrete ones, and moreover when T which are insuiciently well-behaved, it is the normalized T -monoids which are o more interest. Thus, we subsequently discard the notion o object-discreteness. (Hermida s generalized multicategories also arise as normalized T -monoids, where X consists o discrete ibrations in a suitable 2-category K and T is an extension o a suitable 2-monad on K. Weber s deinition is a special case, since as given it really only makes sense or generalized

6 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 585 operads, or which normalization is automatic; see.16.) In Table 1 we summarize the meanings o T -monoids and normalized T -monoids or a number o monads T. Now, what determines whether the right notion o generalized multicategory is a plain T -monoid or a normalized one? The obvious thing distinguishing the situations o Leinster, urroni, and Clementino et. al. rom those o aez-dolan, Cheng, and Hermida is that in the ormer case, the objects o X are set-like, whereas in the latter, they are category-like. However, some types o generalized multicategory can be constructed starting rom two dierent monads on two dierent virtual equipments, one o which belongs to the irst group and the other to the second. For example, observe that an ordinary (non-symmetric) multicategory has an underlying ordinary category, containing the same objects but only the morphisms (V 1 ) W with unary source. Thus, such a multicategory can be deined in two ways: either as extra structure on its set o objects, or as extra structure on its underlying category. In the second case, normalization is the requirement that in the extra added structure, the multimorphisms with unary source do no more than reproduce the originally given category. Thus, ordinary multicategories arise both as T -monoids the ree monoid monad on sets and spans, and as normalized T -monoids or the ree monoidal category monad on categories and prounctors. Our second result in 8 is a generalization o this relationship. We observe that the virtual equipment o categories and prounctors results rom applying the monoids and modules construction Mod to the virtual equipment o sets and spans. Thus, we generalize this situation by showing that or any monad T on a virtual equipment, plain T -monoids can be identiied with normalized Mod(T )-monoids. That this is so in the examples can be seen by inspection o Table 1. Moreover, it is sensible because application o Mod takes set-like things to category-like things. It ollows that the notion o normalized T -monoid is actually more general than the notion o T -monoid, since arbitrary T -monoids or some T can be identiied with the normalized S-monoids or some S (namely S = Mod(T )), whereas normalized S-monoids cannot always be identiied with the arbitrary T -monoids or any T. (For instance, this is not the case when S is the ree symmetric monoidal category monad on categories and prounctors.) This motivates us to claim that the right notion o generalized multicategory is a normalized T -monoid, or some monad T on a virtual equipment. Having reached this conclusion, we also take the opportunity to propose a new naming system or generalized multicategories which we eel is more convenient and descriptive. Namely, i (pseudo) T -algebras are called widgets, then we propose to call normalized T -monoids virtual widgets. The term virtual double category is o course a special case o this: virtual double categories are the normalized T -monoids or the monad T on Pro(Grph) whose algebras are double categories. O course, virtual used in this way is a red herring adjective 1 akin to pseudo and lax, since a virtual widget is not a widget. The converse, however, is true: every widget 1 The mathematical red herring principle states that an object called a red herring need not, in general, be either red or a herring.

7 586 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN Monad T on T -monoid Normalized T -Monoid Pseudo T -algebra Id V-Mat V-enriched Set Set category Id Span(C) Internal category Object o C Object o C in C Id Rel Ordered Set Set Set Id R + -Mat Metric Space Set Set Powerset Rel Closure Space T 1 Closure Space Complete Semilattice Mod(powerset) 2-Pro Modular Closure Closure Space Meet-Complete Space Preorder Ultrailter Rel Topological Space T 1 space Compact Hausdor space Mod(ultrailter) 2-Pro Modular Top. Space Topological Space Ordered Compact Hausdor space Ultrailter R + -Mat Approach space? Compact Hausdor space Free monoid Span(Set) Multicategory? Monoid Mod(ree monoid) Set-Pro Enhanced multicategory Multicategory Monoidal category Free sym. strict Set-Pro Enhanced sym. Symmetric Symmetric mon. mon. cat. multicategory multicategory cat. Free category Span(Grph) Virtual double category? Category Mod(ree category) Pro(Grph)? Virtual double Pseudo double category category Free cat. w/ Set-Pro? Multi-sorted Cat. w/ inite inite products Lawvere theory products Free cat. w/ Set-Pro ls? Monad on Set Cat. w/ small small products ( products Free preshea Span Set ob(s)) Functor A S Functor A S Functor S op Set w/ discrete ibers S op Set Mod(ree preshea) Pro(Set ob(s) )? Functor A S Pseudounctor S op Cat Free strict Span(Globset) Multi-sorted? Strict ω-category ω-category globular operad Mod(ree ω-cat.) Pro(Globset)? Multi-sorted globular operad Monoidal globular cat. Span(Set)? M-set Free M-set (M a monoid) M-graded category Table 1: Examples o generalized multicategories. The boxes marked? do not have any established name; in most cases they also do not seem very interesting.

8 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 587 has an underlying virtual widget, so the terminology makes some sense. For example, the observation above that every monoidal category has an underlying multicategory is an instance o this act. Moreover, it oten happens that virtual widgets share many o the same properties as widgets, and many theorems about widgets can easily be extended to virtual widgets. Thus, it is advantageous to use a terminology which stresses the close connection between the two. Another signiicant advantage o virtual widget over T - multicategory is that requently one encounters monads T or which T -algebras have a common name, such as double category or symmetric monoidal category, but T itsel has no name less cumbersome than the ree double category monad or the ree symmetric monoidal category monad. Thus, it makes more sense to name generalized multicategories ater the algebras or the monad than ater the monad itsel. y the end o 8, thereore, we have uniied all existing notions o generalized multicategory under the umbrella o virtual T -algebras, where T is a monad on some virtual equipment. Since getting to this point already takes us over 40 pages, we leave to uture work most o the development o the theory and its applications, along with more speciic comparisons between existing theories (see [CS10a, CS10b]). However, we do spend some time in 9 on the topic o representability. This is a central idea in the theory o generalized multicategories, which states that any pseudo T -algebra (or, in act, any oplax T -algebra) has an underlying virtual T -algebra. Additionally, one can characterize the virtual T -algebras which arise in this way by a representability property. This can then be interpreted as an alternate deinition o pseudo T -algebra which replaces coherent algebraic structure by a universal property, as advertised in [Her01]. In addition to the identiication o monoidal categories with representable multicategories, this also includes the act that compact Hausdor spaces are T 1 spaces with additional properties, and that ibrations over a category S are equivalent to pseudounctors S op Cat. In [CS10b] we will extend more o the theory o representability in [Her01] to our general context. Finally, the appendices are devoted to showing that all existing notions o generalized multicategory are included in our ramework. In Appendix A we prepare the way by giving suicient conditions or our constructions on virtual double categories to preserve composites, which is important since most existing approaches use bicategories. Then in Appendix we summarize how each existing theory we are aware o its into our context. We have chosen to postpone these comparisons to the end, so that the main body o the paper can present a uniied picture o the subject, in a way which is suitable also as an introduction or a reader unamiliar with any o the existing approaches. It should be noted, though, that we claim no originality or any o the examples or applications, or the ideas o representability in 9. Our goal is to show that all o these examples all into the same ramework, and that this general ramework allows or a cleaner development o the theory Acknowledgements. The irst author would like to thank ob Paré or his suggestion to consider double triples, as well as helpul discussions with Maria Manuel Clementino, Dirk Homann, and Walter Tholen. The second author would like to thank

9 588 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN David Carchedi and the patrons o the n-category Caé blog or several helpul conversations. oth authors would like to thank the editor and the reeree or helpul suggestions. 2. Virtual double categories The deinition o virtual double category may be somewhat imposing, so we begin with some motivation that will hopeully make it seem inevitable. We seek a ramework which includes all sorts o generalized multicategories. Since categories themselves are a particular sort o generalized multicategory (relative to an identity monad), our ramework should in particular include all sorts o generalized categories. In particular, it should include both categories enriched in a monoidal category V and categories internal to a category C with pullbacks, so let us begin by considering how to uniy these two situations. We start by recalling that both V-enriched categories and C-internal categories are particular cases o monoids in a monoidal category. On the one hand, i V is a cocomplete closed monoidal category and O is a ixed set, then V-enriched categories with object set O can be identiied with monoids in the monoidal category o O-graphs in V i.e. (O O)-indexed amilies o objects o V, with monoidal structure given by matrix multiplication. On the other hand, i C is a category with pullbacks and O is an object o C, then C-internal categories with object-o-objects O can be identiied with monoids in the monoidal category o O-spans in C i.e. diagrams o the orm O A O, with monoidal structure given by span composition. Now, both o these examples share the same deect: they require us to ix the objects (the set O or object O). In particular, the morphisms o monoids in these monoidal categories are unctors which are the identity on objects. It is well-known that one can eliminate this ixing o objects by combining all the monoidal categories o graphs and spans, respectively, into a bicategory. (In essense, this observation dates all the way back to [én67].) In the irst case the relevant bicategory V-Mat consists o V-matrices: its objects are sets, its arrows rom X to Y are (X Y )-matrices o objects in V, and its composition is by matrix multiplication. In the second case the relevant bicategory Span(C) consists o C-spans: its objects are objects o C, its arrows rom X to Y are spans X A Y in C, and its composition is by pullback. It is easy to deine monoids in a bicategory to generalize monoids in a monoidal category 2. However, we still have the problem o unctors. There is no way to deine morphisms between monoids in a bicategory so as to recapture the correct notions o enriched and internal unctors in V-Mat and Span(C). ut we can solve this problem i instead o bicategories we use (pseudo) double categories, which come with objects, two dierent 2 Monoids in a bicategory are usually called monads. However, we avoid that term or these sorts o monoids or two reasons. Firstly, the morphisms o monoids we are interested in are not the same as the usual morphisms o monads (although they are related; see [LS02, ]). Secondly, there is potential or conusion with the monads on bicategories and related structures which play an essential role in the theory we present.

10 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 589 kinds o arrow called horizontal and vertical, and 2-cells in the orm o a square: oth V-Mat and Span(V) naturally enlarge to pseudo double categories, interpreting their existing arrows and composition as horizontal and adding new vertical arrows. For V-Mat the new vertical arrows are unctions between sets, while or Span(C) the new vertical arrows are morphisms in C. We can now deine monoids in a double category (relative to the horizontal structure) and morphisms between such monoids (making use o the vertical arrows) so as to recapture the correct notion o unctor in both cases (see Deinition 2.8). The inal generalization rom pseudo double categories to virtual double categories is more diicult to motivate at the moment, but as remarked in the introduction, we will ind it essential in 4. Conceptually (and, in act, ormally), a virtual double category is related to a pseudo double category in the same way that a multicategory is related to a monoidal category. Thus, just as one can deine monoids in any multicategory, one can likewise do so in any virtual double category Deinition. A virtual double category X consists o the ollowing data. A category X (the objects and vertical arrows), with the arrows written vertically: X Y For any two objects X, Y X, a class o horizontal arrows, written horizontally with a slash through the arrow: X Cells, with vertical source and target, and horizontal multi-source and target, written as ollows: p 1 X 0 p 2 X X11 p 3 X X2 2 p n X Xn n Y 0 Y 1 p q Y g (2.2) α Note that this includes cells with source o length 0, in which case we must have X 0 = X n ; such cells are visually represented as ollows: Y 0 X α g Y 0 Y 1 q

11 590 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN For the ollowing coniguration o cells, X 0 p 11...p 1n1 Xn1 X p 21...p 2n1 Xn2 X X n2 X nm 0 α α α 3 α m Y 0 q 1 Y 1 q 2 Y 2 q 3 q m Y m g 0 β Z 0 Z 1 a composite cell X 0 p 11...p 1n1 Xn1 X r 0 Y 0 β(α m α 1 ) Y m g 0 Z 0 Z 1 For each horizontal arrow p, an identity cell r Y m m p 21...p 2n1 X Xn2 X nm g 1 Y m m g 1 X X p 1 p p Y Y Associativity and identity axioms or cell composition. The associativity axiom states that ( ) ( ) γ(β m β 1 ) (α mkm α 11 ) = γ β m (α mkm α m1 ) β 1 (α 1k1 α 11 ) while the identity axioms state that α(1 p1 1 pn ) = α and 1 q (α 1 ) = α 1 whenever these equations make sense Remark. As mentioned in the introduction, virtual double categories have also been called c-multicategories by Leinster [Lei04] and multicatégories by urroni [ur71]. Our terminology is chosen to emphasize their close relationship with double categories, and to it into the general naming scheme o Remark. In much o the double-category literature, it is common or the slashed arrows (spans, prounctors, etc.) to be the vertical arrows. We have chosen the opposite convention purely or economy o space: the cells in a virtual double category it more conveniently on a page when their multi-source is drawn horizontally.

12 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES Examples. As suggested by the discussion at the beginning o this section, monoidal categories, bicategories, 2-categories, multicategories, and pseudo double categories can each be regarded as examples o virtual double categories, by trivializing the vertical or horizontal structure in various ways; see [Lei02, p. 4] or [Lei04, 5.1] or details. We now present the two virtual double categories that will serve as initial inputs or most our examples: spans and matrices. (oth are also described in [Lei04, Ch. 5].) For consistency, we name all o our virtual double categories by their horizontal arrows, rather than their vertical arrows or objects Example. Let (V,, I) be a monoidal category. The virtual double category V-Mat is deined as ollows: its objects are sets, its vertical arrows are unctions, its p horizontal arrows X Y are amilies {p(y, x)} x X,y Y o objects o V (i.e. (X Y )- matrices), and a cell o the orm (2.2) consists o a amily o V-arrows p 1 (x 1, x 0 ) p 2 (x 2, x 1 ) p n (x n, x n 1 ) α q(x n, gx 0 ), one or each tuple (x 0,..., x n ) X 0 X n. When n = 0, o course, the n-ary tensor product on the let is to be interpreted as the unit object o V. In particular, i V is the 2-element chain 0 1, with given by, then the horizontal arrows o V-Mat are relations. In this case we denote V-Mat by Rel. It is well-known that V-matrices orm a bicategory (and, in act, a pseudo double category) as long as V has coproducts preserved by. However, i we merely want a virtual double category, we see that this requirement is unnecessary. (In act, V could be merely a multicategory itsel.) 2.7. Example. For a category C with pullbacks, the virtual double category Span(C) is deined as ollows: its objects and vertical arrows are those o C, its horizontal arrows p X Y are spans X P Y, and a cell o the orm (2.2) is a morphism o spans P 1 X1 P 2 X2 Xn 1 P n α Q lying over and g. When n = 0, the n-ary span composite in the domain is to be interpreted as the identity span X 0 X 0 X 0. Note that in this case, we do need to require that C have pullbacks. I C does not have pullbacks, a more natural setting would be to consider Span(C) as a co-virtual double category, in which the horizontal target o a cell is a string o horizontal arrows. However, co-virtual double categories do not provide the structure necessary to deine generalized multicategories. We now recall the construction o monoids and modules in a virtual double category rom [Lei04, 5.3].

13 592 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN 2.8. Deinition. Let X be a virtual double category. The virtual double category Mod(X) has the ollowing components: The objects (monoids) consist o our parts (X 0, X, x, ˆx): an object X 0 o X, a horizontal endo-arrow X X 0 X 0 in X, and multiplication and unit cells X X 0 X X 0 X 0 x and X 0 X 0 X X 0 X 0 ˆx X 0 X 0 X X 0 satisying associativity and identity axioms. The vertical arrows (monoid homomorphisms) consist o two parts ( 0, ): a 0 vertical arrow X 0 Y0 in X and a cell in X: X X 0 X 0 0 Y 0 Y 0 which is compatible with the multiplication and units o X and Y. The horizontal arrows (modules) consist o three parts (p, p r, p l ): a horizontal arrow p Y 0 in X and two cells in X: X 0 Y 0 X p X 0 X 0 Y 0 p r and X 0 Y 0 p p X 0 Y Y 0 Y 0 p l X 0 Y 0 p which are compatible with the multiplication and units o X and Y. The cells are cells in X: (X 0 p 1 ) 0 (X 1 p 2 ) 0 (X 2 p 3 ) 0 p n (X n ) 0 0 α (Y 0 ) 0 (Y 1 ) 0 which are compatible with the let and right actions o the horizontal cells. Note that, as observed in [Lei04, 5.3], we can deine Mod(X) without requiring any hypotheses on the virtual double category X, in contrast to the situation or bicategories or pseudo double categories. q g 0

14 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES Example. We denote the virtual double category Mod(V-Mat) by V-Pro; its objects are V-enriched categories, its vertical arrows are V-unctors, its horizontal arrows are V-prounctors, and its cells are a generalization o the orms o [DS97] (including, as a special case, natural transormations between prounctors). When V is closed (hence enriched over itsel) and symmetric, V-prounctors C H D can be identiied with V- unctors D op C V. Again, note that because we are working with virtual double categories, we do not require that V have any colimits (in act, V could be merely a multicategory) Example. Let C be a category with pullbacks. We denote the virtual double category Mod(Span(C)) by Pro(C); it consists o internal categories, unctors, prounctors, and transormations in C. Note that Set-Mat = Span(Set) and thus Set-Pro = Pro(Set). 3. Monads on a virtual double category We claimed in 1 that the inputs o a generalized multicategory are parametrized by a monad. Why should this be so? Suppose that we have an operation T which, given a set (or object) o objects X, produces a set (or object) T X intended to parametrize such inputs. For ordinary multicategories, T X will be the set o inite lists o elements o X. Now, rom the perspective o the previous section, the data o a category includes an object A 0 and a horizontal arrow A 0 A A 0 in some virtual double category. For example, i we work in Set-Mat, then A is a matrix consisting o the hom-sets A(x, y) or every x, y A 0. Now, instead, we want to have hom-sets A(x, y) whose domain x is an element o T A 0. Thus, it makes sense to consider a horizontal arrow A A 0 T A 0 as part o the data o a T -multicategory. (We use A 0 T A 0 rather than T A 0 A 0, or consistency with Examples 2.9 and 2.10: the codomain o the horizontal arrow datum o a monoid speciies the domains o the arrows in that monoid.) However, we now need to speciy the units and composition o our generalized multicategory. The unit should be a cell into A with 0-length domain, but its source and target vertical arrows can no longer both be identities because A 0 T A 0. In an ordinary 1 multicategory, the identities are morphisms (x) x x whose domain is a singleton list; in terms o Set-Mat this can be described by a cell A 0 A 0 A 0 T A 0 A η where A 0 η T A 0 is the inclusion o singleton lists.

15 594 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN Regarding composition, in an ordinary multicategory we can compose a morphism g (y 1,..., y n ) z not with a single morphism, but with a list o morphisms ( 1,..., n ) where (x i1,..., x iki ) i yi. In terms o Set-Mat this represents the act that we cannot ask or a multiplication cell with domain A A, since the domain o A does not match its codomain, but instead we can consider a cell with domain A T A, where we extend T to act on Set-matrices in the obvious way. Now, however, the codomain o T A is T 2 A 0 ; in order to have a cell with codomain A we need to remove parentheses rom the resulting domain ((x 11,..., x 1k1 ),..., (x n1,..., x nkn )) to obtain a single list. Thus the composition should be a cell A A 0 T A 0 T 2A 0 A 0 T A 0 A where µ is the remove parentheses unction. O course, these unctions η and µ are the structure maps o the ree monoid monad on Set. Thus we see that in order to deine ordinary multicategories, what we require is an extension o this monad to Set-Mat. In order to have a good notion o a monad on a virtual double category, we need at least a 2-category o virtual double categories. Since virtual double categories are themselves a special case o generalized multicategories, it suices to observe that generalized multicategories o any sort orm a 2-category. However, since we have not yet deined generalized multicategories in our context, at this point in our exposition it is appropriate to give an explicit description o the 2-category vdbl. O course, the objects o v Dbl are virtual double categories, and its 1-morphisms (called unctors o virtual double categories) are the obvious structure-preserving maps: unctions rom the objects, vertical and horizontal arrows, and cells o the domain to those o the codomain, preserving all types o source, target, identities, and composition. However, the deinition o 2-morphisms in v Dbl is slightly less obvious Deinition. Given unctors X F Y o virtual double categories, a transormation F θ G consists o the ollowing data. G For each object X in X, a vertical arrow F X θ X GX, which orm the components o a natural transormation between the vertical parts o F and G. T A µ For each horizontal arrow X p Y in X, a cell in Y: F X θ X GX F p F Y θ p θ Y (3.2) GY Gp

16 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 595 An axiom asserting that θ is cell-natural, meaning that whenever this makes sense. θ q (F α) = (Gα)(θ p1 θ pn ) Virtual double categories, unctors, and transormations orm a 2-category denoted v Dbl Example. Any lax monoidal unctor V N W N induces unctors V-Mat W-Mat N and V-Pro W-Pro in an evident way. Moreover, any monoidal natural transormation N ψ M between lax monoidal unctors induces transormations N ψ M in both cases. In this way ( )-Mat and ( )-Pro become 2-unctors rom the 2-category o monoidal categories to vdbl Example. Similarly, any pullback-preserving unctor C N D between categories N with pullbacks induces unctors Span(C) N Span(D) and Pro(C) Pro(D), and any natural transormation N ψ M between such unctors induces transormations N ψ M, thereby making Span( ) and Pro( ) into 2-unctors as well Example. When restricted to bicategories or pseudo double categories, unctors o virtual double categories are equivalent to the usual notions o lax unctor. This is a special case o a general act; see Theorem When transormations o virtual double categories are similarly restricted, they become icons in the sense o [Lac10] (or bicategories) and vertical transormations (or pseudo double categories). y a monad on a virtual double category X, we will mean a monad in the 2-category vdbl. Thus, it consists o a unctor T : X X and transormations η : Id T and µ: T T T satisying the usual axioms. We now give the examples o such monads that we will be interested in Example. Since 2-unctors preserve monads, any pullback-preserving monad on a category C with pullbacks induces monads on Span(C) and Pro(C). Examples o such monads include the ollowing. The ree monoid monad on Set (or, more generally, on any countably lextensive category). The ree M-set monad (M ) on Set, or any monoid M (or more generally, or any monoid object in a category with inite limits). The monad ( ) + 1 on any lextensive category. The ree category monad on the category o directed graphs. The ree strict ω-category monad on the category o globular sets.

17 596 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN Many more examples can be ound in [Lei04, pp ]; see also.1. y the argument above, each o these monads extends to a monad on a virtual double category o spans. The assignments V V-Mat and V V-Pro are also 2-unctorial, but the monads we obtain in this way rom monads on monoidal categories are not usually interesting or deining multicategories. However, there are some general ways to construct monads on virtual double categories o matrices, at least when V is a preorder. The ollowing is due to [Sea05], which in turn expands on [CHT04] Example. y a quantale we mean a closed symmetric monoidal complete lattice. A quantale is completely distributive i or any b V we have b = {a a b}, where a b means that whenever b S then there is an s S with a s. (I in this deinition S is required to be directed, we obtain the weaker notion o a continuous lattice.) For us, the two most important completely distributive quantales are the ollowing. The two-element chain 2 = (0 1). The extended nonnegative reals R + = [0, ] with the reverse o the usual ordering and = +. A unctor said to be taut i it preserves pullbacks o monomorphisms (and thereore also preserves monomorphisms). A monad is taut i its unctor part is taut, and moreover the naturality squares o η and µ or any monomorphism are pullbacks. Some important taut monads on Set are the identity monad, the powerset monad (whose algebras are complete lattices), the ilter monad, and the ultrailter monad (whose algebras are compact Hausdor spaces). Now let V be a completely distributive quantale and T a taut monad on Set. For a V-matrix X where p Y and elements F T X and G T Y, deine T p(g, F) = {v V p v [] = Y : ( G T F T (p v []) )}, { x X } y : v p(y, x). It is proven in [Sea05] that this action on horizontal arrows extends T to a monad on V-Mat. (Actually, Seal shows that it is a lax extension o T to V-Mat with op-lax unit and counit ; we will show in.6 that this is the same as a monad on V-Mat.) In [Sea05] this monad on V-Mat is called the canonical extension o T (note, however, that it is written backwards rom his deinition, as our Kleisli arrows will be X T Y, whereas his are T X Y ). Since V-Mat is isomorphic to its horizontal opposite, there is also an op-canonical extension, which is in general distinct (although in some cases, such as or the ultrailter monad, the two are identical). There are also many other extensions: or more detail, see [SS08]. Another general way o constructing monads on virtual double categories is to apply the construction Mod rom the previous section, which turns out to be a 2-unctor. Its 1- unctoriality is airly obvious and was described in [Lei04, 5.3]; its action on 2-morphisms is given as ollows.

18 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES Deinition. Let X F Y be unctors between virtual double categories, and F θ G a transormation. One can deine a transormation G Mod(F ) Mod(θ) Mod(G) whose vertical-arrow component at an object (X 0, X, x, ˆx) is the monoid homomorphism F (X) F X 0 F X 0 θ X0 θ X0 θ X GX 0 GX 0 G(X) and whose cell component at a horizontal arrow (p, p r, p l ) is given by F p F X 0 F Y 0 θ X0 θ Y0 θ p GX 0 GY 0 Gp 3.9. Proposition. With action on objects, 1-cells, and 2-cells described above, Mod is an endo-2-unctor o vdbl. Note that the 2-unctors ( )-Pro and Pro( ) can now be seen as the composites o Mod with ( )-Mat and Span( ), respectively Corollary. A monad T on a virtual double category X induces a monad Mod(T ) on Mod(X) Example. Any monad T on V-Mat induces a monad on V-Pro. For instance, this applies to the monads constructed in Example Example. Let V be a symmetric monoidal category with an initial object preserved by. Then the ree monoid monad T on Set extends to a monad on V-Mat p as ollows: a V-matrix X T Y is sent to the matrix T X p T Y deined by { ( ) p(y 1, x 1 )... p(y n, x n ) i n = m T p (y 1,..., y m ), (x 1,..., x n ) = i n m Applying Mod, we obtain an extension o the ree strict monoidal V-category monad rom V-Cat to V-Pro.

19 598 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN Example. Likewise, any monad T on Span(C) extends to a monad on Pro(C). ut most interesting monads on Span(C) are induced rom C, so this gains us little beyond the observation that Pro( ) is a 2-unctor. Not every monad on V-Pro or Pro(C) is induced by one on V-Mat or Span(C), however. The ollowing examples are also important Example. Let V be a symmetric monoidal category with inite colimits preserved by on both sides. Then there is a ree symmetric strict monoidal V-category monad T on V-Cat, deined by letting the objects o T X be inite lists o objects o X, with ( ) T X (x 1,..., x n ), (y 1,..., y m ) = σ S n 1 i n X(x σ(i), y i ) i n = m i n m. A nearly identical-looking deinition or prounctors extends this T to a monad on V-Pro. A similar deinition applies or braided monoidal V-categories Example. For V as in Example 3.14, there is also a ree V-category with strictly associative inite products monad on V-Cat. The objects o this T X are again inite lists o objects o X, but now we have ( ) T X (x 1,..., x m ), (y 1,..., y n ) = X(x j, y i ). 1 i n 1 j m I V is cartesian monoidal, then this can equivalently be written as ( ) T X (x 1,..., x m ), (y 1,..., y n ) = X(x α(i), y i ). α: n m 1 i n Again, a nearly identical deinition or prounctors extends this to a monad on V-Pro Example. Monads that reely adjoin other types o limits and colimits also extend rom V-Cat to V-Pro in a similar way. For instance, i V is a locally initely presentable closed monoidal category as in [Kel82], there is a ree V-category with cotensors by initely presentable objects monad on V-Cat. An object o T X consists o a pair 3 (v; x) where x X and v V is initely presentable. On homs we have T X ( (v; x), (w; y) ) = [ w, X(x, y) v ]. As beore, a nearly identical deinition extends this to V-Pro. 3 To be precise, this deinition only gives a pseudomonad on V-Cat. It is, however, easy to modiy it to make a strict monad.

20 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES Generalized multicategories We now lack only one inal ingredient or the deinition o generalized multicategories. Since multicategories are like categories, we expect them to also be monoids in some virtual double category. However, as we have seen in 3, their underlying data should include a horizontal arrow A 0 T A 0 rather than A 0 A 0. Thus we need to construct, given T and X, a virtual double category in which the horizontal arrows are horizontal arrows o the orm A 0 T A 0 in X. This is the purpose o the ollowing deinition Deinition. Let T be a monad on a virtual double category X. Deine the horizontal Kleisli virtual double category o T, H-Kl(X, T ), as ollows. Its vertical category is the same as that o X. A horizontal arrow X p Y is a horizontal arrow X p T Y in X. A cell with nullary source uses the unit o the monad, so that a cell Y X α p g Z in H-Kl(X, T ) is a cell Y X η T X α in X (note that T g η = η g by naturality). A cell with non-nullary source uses the multiplication o the monad, so that a cell X 0 p T g T Z p 1 X 0 p 2 X X11 p 3 X X2 2 Xn p n X n g α Y 0 Y 1 q Y 1

21 600 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN in H-Kl(X, T ) is a cell p 1 X 0 T p 2 T X X11 T 2 T X 2 p 3 T 2 n 1 p n T n X n α T g Y 0 T Y 1 in X (note that T g µ n 1 = µ n 1 T n g, by naturality). The composite o q. µ µ T X n α 1 α 2 α n β is given by the composite o α 1 µ µ µ µ µ µ T µ T α 2 T 2 α 3 µ µ T µ β µ µ T µ T 2 µ T 3 α 4 µ µ T µ T 2 µ µ µ T µ µ µ T µ T n 2 µ T n 2 µ T n 1 α n in X. Identity cells use those o X: X p T Y X 1 P p T Y

22 A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES 601 In general, the associativity or H-Kl(X, T ) is shown by using the (cell) naturality o µ and η, as well as the monad axioms. The general associativity is too large a diagram to show here; instead, we will demonstrate a sample associativity calculation, which is representative o the general situation. Consider the ollowing cells in H-Kl(X, T ): α 1 α 2 α 3 α 4 β 1 β 2 There are two possible ways to compose these cells: either composing the bottom irst: γ (γ(β 1 β 2 ))(α 1 α 2 α 3 α 4 ) or the top two irst, ollowed by composition with the bottom: γ((β 1 (α 1 α 2 )) (β 2 (α 3 α 4 ))) The irst composite is given by the ollowing composite in X: T µ T µ α 1 T α 2 T 2 α 3 T 3 α 4 µ µ β 1 T β 2 y using cell naturality o µ twice, the above becomes γ T µ T µ α 1 T α 2 µ µ β T α 3 1 T 2 α 4 T β 2 γ

23 602 G.S.H. CRUTTWELL AND MICHAEL A. SHULMAN we then use the monad axiom T µ µ = µ µ to get µ µ α 1 T α 2 µ µ β T α 3 1 T 2 α 4 T β 2 γ which is the second composite γ((β 1 (α 1 α 2 )) (β 2 (α 3 α 4 ))) Remark. Unortunately, we do not know o any universal property satisied by this construction. In particular, H-Kl(X, T ) is not a Kleisli object or T in vdbl in the sense o [Str72a]; the latter would instead contain vertical Kleisli arrows. In act, or general X there need not even be a canonical unctor X H-Kl(X, T ). We can now give our irst preliminary deinition o generalized multicategories relative to a monad T Deinition. Let T be a monad on a virtual double category X. A T -monoid is deined to be a monoid in H-Kl(X, T ), and likewise or a T -monoid homomorphism. We denote the virtual double category Mod(H-Kl(X, T )), whose objects are T -monoids, by KMod(X, T ). As a reerence, the data or a T -monoid consists o an object X 0 X, a horizontal arrow X X 0 T X 0 in X, and cells X X 0 T T X X 0 T 2 X 0 µ x and X 0 T X 0 X X 0 η ˆx X 0 T X 0 X Note that these cells have precisely the orms we predicted at the beginning o Remark. We have seen in 2 that Mod is a 2-unctor. In act, under suitable hypotheses (involving the notions o restriction and composites to be introduced in 5 and 7), H-Kl is also a (pseudo) unctor, and thus so is KMod. In act, H-Kl is a pseudo unctor in two dierent ways, corresponding to the two dierent kinds o morphisms o monads: lax and colax. This was observed by [Lei04] in his context; we will discuss the unctoriality o H-Kl and KMod in our ramework in the orthcoming [CS10a]. We now consider some examples.